Dynamic modeling of butterfly subdivision surfaces

In this paper, we develop integrated techniques that unify physics-based mo deling with geometric subdivision methodology and present a scheme for dyna mic manipulation of the smooth limit surface generated by the (modified) bu tterfly scheme using physics-based "force" tools. This procedure-based surf ace model obtained through butterfly subdivision does not have a closed-for m analytic formulation (unlike other well-known spline-based models) and, h ence, poses challenging problems to incorporate mass and damping distributi ons, internal deformation energy, forces, and other physical quantities req uired to develop a physics-based model. Our primary contributions to comput er graphics and geometric modeling include: 1) a new hierarchical formulati on for locally parameterizing the butterfly subdivision surface over its in itial control polyhedron, 2) formulation of dynamic butterfly subdivision s urface as a set of novel finite elements, and 3) approximation of this new type of finite elements by a collection of existing finite elements subject to implicit geometric constraints. Our new physics-based model can be scul pted directly by applying synthesized forces and its equilibrium is charact erized by the minimum of a deformation energy subject to the imposed constr aints. We demonstrate that this novel dynamic framework not only provides a direct and natural means of manipulating geometric shapes, but also facili tates hierarchical shape and nonrigid motion estimation from large range an d volumetric data sets using very few degrees of freedom (control vertices that define the initial polyhedron).